I have done some further testing and I have a theory that this was either a bug or a disabled / missing feature:

8.1.2 / 8.2.1

Cannot enable MGMT interface no matter what I try

8.2.5+ / 8.3.0 / 8.4.0

Can enable the MGMT (eth0) interface with no problems at all, 100% repeatable

It appears that not being able to enable the MGMT interface on AWS was indeed a bug up that was fixed sometime after 8.2.1.

I was not able to find any workaround (e.g. upgrading through various versions from different starting points, all the UI config tricks in the world, etc) other than upgrading to a version that allows the MGMT interface to be enabled and operational. I used at least 8.2.5 and was able to get MGMT to work reliably everytime I wanted to enable it.